Laureano González Vega

On algorithms for real algebraic plane curves

During the last years several researchers have considered the problem of finding polynomial-time sequential algorithms for the computation of the topology of a real algebraic plane curve. Up till now, we can divide these algorithms into two main groups: those coding real algebraic numbers by means of isolating intervals (see for instance [3] and [7]) and those coding them a la Thom (see [9]). The aim of this note is to survey the state of the art as far as the second group is concerned. In particular we introduce two new such algorithms, one of which is, to our knowledge, the algorithm computing the topology of a real algebraic plane curve with the lowest running time, while the other one has been successfully implemented.

Various New Expressions for Subresultants and Their Applications

Abstract.This article is devoted to presenting new expressions for Subresultant Polynomials, written in terms of some minors of matrices different from the Sylvester matrix. Moreover, via these expressions, we provide new proofs for formulas which associate the Subresultant polynomials and the roots of the two polynomials. By one hand, we present a new proof for the formula introduced by J. J. Sylvester in 1839, formula written in terms of a single sum over the roots. By other hand, we introduce a new expression in terms of the roots by considering the Newton basis.

Hilbert stratification and parametric gröbner bases

In this paper we generalize a method to analyze inhomogeneous polynomial systems containing parameters. In particular, the Hilbert function is used as a tool to check that the specialization of a “generic” Grobner basis of the parametric polynomial system (computed in a polynomial ring having both parameters and unknowns as variables) is a Grobner basis of the specialized system. Extending the analysis, we can also build the so-called Hilbert stratification of the associated variety. We classify the possible specializations according to the value of the Hilbert function of the specialized system. Some computation examples with the PoSSoLib are reported.

Continuity properties for flat families of polynomials (I) Continuous parametrizations

Abstract Inspired by classical results in algebraic geometry, we study the continuity with respect to the coefficients, of the zero set of a system of complex homogeneous polynomials with a given pattern and when the Hilbert polynomial of the generated ideal is fixed. In this work we prove topological properties of some classifying spaces, e.g. the space of systems with given pattern, fixed Hilbert polynomial is locally compact, and we establish continuous parametrizations of Nullstellensatz formulae. In the general case we get local rational results but in the complex case we get global results using rational polynomials in the real and imaginary parts of the coefficients. In a second companion paper, we shall treat the continuity of zero sets for the Hausdorff distance, i.e., from a metric point of view.

Computing the Topology of an Arrangement of Implicit and Parametric Curves Given by Values

Curve arrangement studying is a subject of great interest in Computational Geometry and CAGD. In our paper, a new method for computing the topology of an arrangement of algebraic plane curves, defined by implicit and parametric equations, is presented. The polynomials appearing in the equations are given in the Lagrange basis, with respect to a suitable set of nodes. Our method is of sweep-line class, and its novelty consists in applying algebra by values for solving systems of two bivariate polynomial equations. Moreover, at our best knowledge, previous works on arrangements of curves consider only implicitly defined curves.